Quasi-Static Variation of Power-Law and Log-Normal Distributions of Urban Population

We analytically derived and confirmed by empirical data the following three relations from the quasi-time-reversal symmetry, Gibrat’s law, and the non-Gibrat’s property observed in the urban population data of France. The first is the relation between the time variation of the power law and the quasi-time-reversal symmetry in the large-scale range of a system that changes quasi-statically. The second is the relation between the time variation of the log-normal distribution and the quasi-time-reversal symmetry in the mid-scale range. The third is the relation among the parameters of log-normal distribution, non-Gibrat’s property, and quasi-time-reversal symmetry.     

Classical versus Bayesian risks in acceptance sampling: a sensitivity analysis

Assuming a beta prior distribution on the fraction defective, $p$ , failure-censored sampling plans for Weibull lifetime models using classical (or average) and Bayesian (or posterior) producer’s and consumer’s risks are designed to determine the acceptability of lots of a given product. The average risk criterion provides a certain assurance that good (bad) lots will be accepted (rejected), whereas the posterior risk criterion provides a determined confidence that an accepted (rejected) lot is indeed good (bad). The performance of classical and Bayesian risks are analyzed in developing sampling plans when the lifetime variable follows the Weibull distribution. Several figures and tables illustrate the sensitivity of the risks and optimal sample sizes for selected censoring levels and specifications according to the available prior information on $p$ . The analysis clarifies the distinction among the different risks for a given sampling plan, and the effect of the prior knowledge on the required sample size. The study shows that, under uncertainty in the prior variance of $p$ , the designs using Bayesian risks are more appropriate.     

Smoothness of the Distribution of the Supremum of a Multi-dimensional Diffusion Process

In this article we deal with a multi-dimensional diffusion whose corresponding diffusion vector fields are commutative, and study its joint distribution at the time when a component attains its maximum on finite time interval. Under regularity and ellipticity conditions we shall show the smoothness of this joint distribution.     

Estimates for the density of functionals of SDEs with irregular drift

We obtain upper and lower bounds for the density of a functional of a diffusion whose drift is bounded and measurable. The argument consists of using Girsanov’s theorem together with an Itô–Taylor expansion of the change of measure. One then applies Malliavin calculus techniques in a non-trivial manner so as to avoid the irregularity of the drift. An integration by parts formula for this set-up is obtained.     

Synthesis and Structure of the Dimeric Copper(II) Complex Tetrakis[<Emphasis Type="Italic">N</Emphasis>-thiazol-2-yl-(4-methylphenyl)sulfonamidate]dicopper(II)

Abstract The coordination chemistry of the ligand N-thiazol-2-yl-toluenesulfonamidate towards the copper(II) ion has been investigated using an electrochemical synthesis method. The X-ray structure of this complex was elucidated and is discussed. The compound crystallised in the monoclinic crystal system, P2₁/c space group with a = 17.3888(9), b = 16.3003(9), c = 18.3679(9) ? and β = 114.3640(10)°. Four bidentate sulfathiazolato anions bridge two metal centers in a paddle-wheel fashion, with the nitrogen atoms as donors to give a dimeric species with a Cu···Cu distance of 2.7859(5) ?. Graphical Abstract The coordination chemistry of the ligand N-thiazol-2-yl-toluenesulfonamidate towards the copper(II) ion has been investigated using an electrochemical synthesis method. The X-ray structure of the product has been elucidated and is discussed. Sulfathiazolato anions act as bridging ligands to give a dimeric species with a Cu···Cu distance of 2.7859(5) ?.      

Numerical investigation of the current transition regimes in nanochannels

The concentration polarization phenomena and its effects represent one of the main challenges for the optimal operation of many nanofluidic systems. A numerical investigation of the different electric current transition regimes observed during the concentration polarization phenomena in nanochannels is performed. This included a 2D‐axisymmetric simulation of the nanofluidic system (reservoir‐nanochannel‐reservoir). From these simulations, a novel mechanism is discovered that explains that different current transition regimes. This driving mechanism involves the applied electric field penetration while the convective flow mechanism is found to be negligible. This differs with the classical statement that the mixing process with less depleted areas initiated by an electrokinetic vortex instability starts the overlimiting regime. Additionally, the numerical approach allows us to identify new characteristics of the linear‐limiting transition such as source‐like and saddle‐like points of the electric field streamlines. The three voltage–current regimes (linear, limiting and overlimiting) are explained by observing and quantifying changes in electric field, potential, ion concentration and ion concentration gradients within the system.     

Correction factors for a total scatter/backscatter nephelometer

Total aerosol scattering and backscattering atmospheric values are typically obtained with an integrating nephelometer. Due to design limitations, measurements do not cover the full (0°–180°) angular range, and correction factors are necessary. The effect of angle cutoff is examined for a number of particle size distributions and refractive indices. Scattering data for sub-micron particles can be corrected by the use of a modified Anderson approximation, while data for larger particle distributions can be approximated by a function of the effective size parameter. Correction factors for the hemispheric backscatter ratio are found to be small if nonsphericity is assumed. Such approximations will help more accurate corrections for angle range, particularly at large size parameter values.     

Unprecedented Ring–Ring Interconversion of N,P,C‐Cage Ligands

The novel N,P,C‐cage complexes 5 a – f and 6 a – f have been obtained by the reaction of the P‐pentamethylcyclopentadienylphosphinidene complex 2 , generated thermally from 2H‐azaphosphirene complex 1 , with N‐methyl‐C‐arylcarbaldimines 3 a – f . Li/Cl phosphinidenoid complex 8 reacted with 3 a , b to give N,P,C‐cage complexes 6 a , b , whereas with 3 c – f , complexes 6 c – f were obtained in negligible amounts only. Both types of ligand N,P,C‐cage structures 5 and 6 were found to be in an unprecedented equilibrium, with 5 a , f as the predominant species. Transient electrophilic terminal phosphinidene complexes 10 a – f serve as intermediates in both ligand interconversions ( 5 a , f ? 6 a , f ), as evidenced through trapping reactions with phenylacetylene and N‐methyl‐C‐phenylcarbaldimine, thus leading to the novel N,P,C‐cage complexes 13 b and 15 . DFT calculations predicted a small difference in the relative energies of the two types of N,P,C‐cage ligands, and a remarkable stabilisation of the aminophosphinidene complex 10 as the common precursor, thereby providing an insight into this surprising 5‐ring–3‐ring interconversion. In depth analysis of intermediate 10 revealed the occurrence of both through‐bond (conventional inductive/mesomeric effects) and through‐space (non‐covalent interactions) mechanisms, which amount to 67.8 and 14.4 kcal mol^?1, respectively, and account for the remarkable stabilisation of this intermediate.